The lecturer nicely summarized the main methods of solving quantum mechanics problems.
- Separable solution: gives you special functions (Hamiltonian has some symmetry)
- Perturbation theory: sometimes works well, but not always
- Semiclassical method: works for large wavelengths
- Numerical analysis: broadly applicable but doesn't give much physical intuition
The first method isn't realistic; we don't have perfect rectangular and spherical potentials. Perturbation theory doesn't work very well in highly degenerate systems (e.g. the high energy levels of a stadium billiard). So for analytic solutions to general problems, we can only use the semiclassical method.
The lecturer's insight was that the generic Hamiltonian in quantum mechanics is partly chaotic and partly quasi-periodic. We can deform a symmetric Hamiltonian or quantum system a small amount and if the system is in certain areas of phase space, the system still has locally conserved quantities (KAM theorem from chaos theory).
This sounds a little bit like Noether's theorem where you make an infinitesimal transformation which leaves the Hamiltonian invariant. Noether's theorem says that for a continuous symmetry like this, there is a corresponding conserved quantity. The KAM theory adds to this by saying that symmetric Hamiltonians can be robust to small perturbations, which break their symmetry.
A friend of mine raised the following objection:
Probably true, but don't forget scattering theory for the continuum, which has lots of powerful methods related to perturbation theory and semiclassics. Also the partial wave expansion of scattering theory is powerful even when the system is not spherically symmetric in the same way that the multipole expansion is useful for arbitrary current and charge distributions when considering radiation and other distant fields.
For fun, here's a Poincare section I made in a graduate level class on mechanics and chaos:
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